GE 331-Lecture 17 - Bivariate Normal Distribution Recall:...

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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 1 • Recall: Normal distribution: • Let R.V.s U and V be independent normal, what can we say about X=aU+bV and Y=cU+dV? (a, b, c, d are scalars) • They are Normal too, but not independent. • In fact any linear combination of X and Y is normal! Bivariate Normal Distribution = 2 2 2 ) ( exp 2 1 ) ( σ μ π x x f
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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 2 Bivariate Normal Distribution • U and V are independent normal, X=aU+bV and Y=cU+dV (a, b, c, d are scalars) X,Y are jointly normal (Gaussian) (Attention jointly normal is not same as normal!) The Joint PDF is called a bivariate normal distribution: with = ) ( ) ( 2 1 exp | | 2 1 ) ( 1 T μ x Σ μ x Σ x π f = = = 2 2 , , y y x y x x Y X y x σ ρσ μ Σ μ x
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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 3 Bivariate Normal Distribution • Normal distribution ( X,Y ): Bivariate normal distribution with = 2 2 2 ) ( exp 2 1 ) ( σ μ π x x f = ) ( ) ( 2 1 exp | | 2 1 ) ( 1 T μ x Σ μ x Σ x f = = = 2 2 , , y y x y x x Y X y x ρσ Σ μ x
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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 4 Bivariate Normal Distribution (Cont.) • If (X,Y) has bivariate normal distribution Marginal probability distribution Covariance and correlation Uncorrelated ( ρ =0 ) <=> independence ) , ( ~ ), , ( ~ 2 2 Y Y X X N Y N X σ μ ρ ρσ = = ) , ( ) , cov( Y X corr Y X Y X
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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 5 • Descriptive statistics organizing, summarizing and displaying data • Inferential statistics using sample data to draw conclusions about a population Statistics
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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 6 • Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect
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This note was uploaded on 09/08/2009 for the course GE 331 taught by Professor Negarkayavash during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE 331-Lecture 17 - Bivariate Normal Distribution Recall:...

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